Dynamic linear economies with social interactions
∗Onur ¨Ozg¨ur Universit´e de Montr´eal†
CIREQ, CIRANO
Alberto Bisin New York University‡
NBER and CIREQ February 15, 2011
Abstract
Social interactions arguably provide a rationale for several important phenomena, from smok-ing and other risky behavior in teens to e.g., peer effects in school performance. We study social interactions in dynamic economies. For these economies, we provide existence (Markov Perfect Equilibrium in pure strategies), ergodicity, and welfare results. Also, we characterize equilibria in terms of agents’ policy function, spatial equilibrium correlations and social mul-tiplier effects, depending on the nature of interactions. Most importantly, we study formally the issue of the identification of social interactions, with special emphasis on the restrictions imposed by dynamic equilibrium conditions.
Journal of Economic LiteratureClassification Numbers: C31, C62, C72, C73, Z13.
Keywords: Conditional covariance stationarity, conformity, ergodicity, habits, identification, local and global interactions, Markov perfect equilibrium, social interactions, social norms, social processes, social status, spatial correlations.
∗
This paper is part of the Polarization and Conflict Project CIT-2-CT-2004-506084 funded by the European Commission-DG Research Sixth Framework Programme. We are grateful to Bernard Salani´e for some insightful comments early on. We also thank Massimiliano Amarante, Jess Benhabib, Ken Binmore, Michele Boldrin, Yann Bramoull´e, Pierre-Andr´e Chiappori, In-Koo Cho, Aureo De Paula, Itzhak Gilboa, Bryan Graham, Ali Horta¸csu, Fr´ed´eric Koessler, Takashi Kunimoto, Justin Leroux, David Levine, Bob Lucas, Andy McLennan, Debraj Ray, Larry Samuelson, Manuel Santos, Tom Sargent, Jos´e Scheinkman, Karl Schlag, David Schmeidler, Paolo Siconolfi, Yves Sprumont, Bruno Strulovici, Jean Tirole, and many seminar participants. Part of this research was done while
¨
Ozg¨ur was visiting the Economics Department at the Universit´e Laval. Thanks to Yann Bramoull´e and Bernard Fortin for organizing the visit. ¨Ozg¨ur is grateful for financial support to “La Chaire du Canada en ´Economie des Politiques Sociales et des Ressources Humaines” at Universit´e Laval, CIREQ, CIRP ´EE, and FQRSC.
†
Department of Economics; e-mail: onur.ozgur@umontreal.ca
‡
1
Introduction
Agents interact in markets as well as socially, that is, in the various socioeconomic groups they belong to. Models of social interactions are designed to capture in a simple abstract way socioeco-nomic environments in which markets do not mediate all of agents’ choices. In such environments agents’ choices are determined by their preferences as well as by their ability to interact with others, on their position in a predetermined network of relationships, e.g., a family, a peer group, or more generally any socioeconomic group.1
Social interactions arguably provide a rationale for several important phenomena, Peer effects, in particular, have been indicated as one of the main empirical determinants of risky behavior in adolescents.2 Relatedly, peer effects have been studied in connection with education outcomes,3
obesity,4 friendship and sex,5labor market referrals,6neighborhood and employment segregation, 7 criminal activity,8 and several other socioeconomic phenomena.9
The large majority of the existing models of social interactions are static; or, when dynamic models of social interactions are studied, it is typically assumed that agents are myopic and their choices are subject to particular behavioral assumptions.10 In this paper, we contribute
1
The integration of models of social interactions within economic theory is an active and interesting area of research. See the recent Handbook of Social Economics, Benhabib, Bisin, and Jackson (2010).
2See e.g., Ali and Dwyer (2009), Axtell et al. (2006), Bauman and Ennett (1996), Bifulco et al. (2009),
Chaloupka and Warner (2000), Clark and Loheac (2007), Cook and Moore (2000), Cutler and Glaeser (2007), DeCicca et al. (2008), Evans et al. (1992), Fletcher (2009), Gaviria and Raphael (2001), Gilleskie and Strumpf (2005), Gilleskie and Zhang (2010), Jones (1984), Kobus (2003), Krauth (2005, 2006), Kremer and Levy (2008), Krosnick and Judd (1982), Lewitt et al. (1981), Lundborg (2006), Nakajima (2007), Norton et al. (1998), Powell et al. (2003), Sacerdote (2001), Soetevent and Kooreman (2007), Tyas and Pederson (1998), Wang et al. (1995, 2000).
3
Altonji et al (2005), Ammermuller and Pischke (2009), B´enabou (1996), Borjas (1995), Boozer and Cacciola (2001), Carrell et al. (2009), De Giorgi et al. (2009), Evans et al. (1992), Gaviria and Raphael (2001), Hoxby (2000a, 2000b), Soetevent and Kooreman (2007), Zimmerman (2003).
4
Burke and Heiland (2007), Christakis and Fowler (2007).
5Akerlof et al. (1996), Bearman et al. (2004), Bramoull´e et al. (2009), Broadhead et al. (1998), Cipollone and
Rosolia (2007), Conti et al. (2009), Currarini et al. (2009), Kandel (1978), Leider et al. (2007), Mihaly (2007), Moody (2001).
6Bayer et al. (2008), Bjorn and Vuong (1985), Calvo-Armengol and Jackson (2004), Conley and Topa (2002,
2007), De Giorgi et al. (2009), Goldin and Katz (2002), Granovetter (1973, 1995), Grodner and Kniesner (2007), Ichino and Falk (2006), Ioannides and Datcher Loury (2004), Moro (2003), Topa (2001), Weinberg et al. (2004).
7
Aizer and Currie (2004), B´enabou (1993), Case and Katz (1991), Crane (1991), Durlauf (1996, 2004), Goering and Feins (1997), Hoff and Sen (2005), Ioannides and Topa (2009), Ioannides and Zabel (2008), Ioannides and Zanella (2009), Katz et al. (2001), Ludwig et al. (2001), Mobius (2000), Rosenbaum (1995), Schelling (1971, 1972).
8
Calvo Armengol et al. (2009), Glaeser et al. (1996), Kling et al. (2005), Ludwig et al. (2001).
9See Bisin et al. (2010), Glaeser and Scheinkman (2001), Moffitt (2001) for surveys. 10
Exceptions include an example on female labor force participation in Glaeser and Scheinkman (2001), Binder and Pesaran (2001) on life-cycle consumption under social Interactions, Blume (2003) on social stigma, Brock and Durlauf (2010) and de Paula (2009) on duration models, and the theoretical analysis of Bisin, Horst, and ¨Ozg¨ur
to this literature by studying social interactions in dynamic economies. We focus our attention on linear economies, in which each agent’s preferences are quadratic. Dynamic linear models of course have appealing analytical properties. Hansen and Sargent (2004) study this class of models systematically, exploiting the tractability of linear control methods and matrix Riccati equations. While the class of economies we study in this paper allows however for a countable number of heterogeneous agents and an infinite horizon, giving rise to infinite dimensional systems, some tractability is maintained. Furthermore, in the class of economies we study agents display preferences for conformity, that is, preferences which incorporate the desire to conform to the choices of agents in a reference group.
More specifically, each agent’s preferences are hit by random preference shocks over time. Each agent interacts with agents in his social reference group, in the sense that each agent’s instantaneous preferences depend on the current choices of agents in his social reference group, as a direct externality. Each agent’s instantaneous preferences also depend on the agent’s own previous choice, representing the inherent costs to dynamic behavioural changes due e.g., to habits. When agents’ reference groups overlap, each agent’s optimal choice depends on all the other agents’s previous choices and current preference shocks, as long as they are observable. We allow for complete and incomplete information with respect to preference shocks. Requiring that the social and informational structure of each agent satisfy a symmetry condition, we restrict our analysis to symmetric Markov perfect equilibria. Agents’ choices at equilibrium are determined by linear policy (best reply) functions. More specifically, e.g., in infinite-horizon economies, a symmetric Markov perfect equilibrium is represented by a symmetric policy function, for each arbitrary agent a ∈ A, a countable set, which maps the agent’s choice at time t, xa
t, linearly in
each agent’s past choices, xa+bt−1, in each agent’s contemporaneous idiosyncratic preference shock, θta+b, and in the mean preference shock, θ:
xat =X
b∈A
cbxa+bt−1 +X
b∈A
dbθta+b+ e θ
For these economies, we provide some fundamental theoretical results: (Markov perfect) equi-libria exist (for finite economies they are unique) and they induce an ergodic stochastic process over the equilibrium configuration of actions. Furthermore, a stationary ergodic distribution exists. We also derive a recursive algorithm to compute equilibria. The proof of the existence theorem, in particular, requires some subtle arguments. In fact, standard variational arguments require to bound the marginal effect of any infinitesimal change dxa on the agent’s value
func-tion. But in the class of economies we study, the Envelope theorem (as e.g., in Benveniste and Scheinkman (1979)) is not sufficient for this purpose, as dxa affects agent a’s value function
di-rectly and indidi-rectly, through its effects on all agents b ∈ A\a’s choices, which in turn affect agent a’s value function. The marginal effect of any infinitesimal change dxa is then an infinite sum of
endogenous terms. In our economy, however, we can exploit the linearity of policy functions to represent a symmetric MPE by a fixed point of a recursive map which can be directly studied.
Exploiting the linear structure of our economies we can study equilibria in some detail, char-acterizing the parameters of the policy function as well as a fundamental statistical property of equilibrium, the cross-sectional auto-correlation of actions. In turn we obtain a series of results regarding the welfare properties of equilibrium and various comparative dynamics exercises of interest. First of all, we show that, since social interactions are modelled in this paper as a preference externality, equilibria will not be efficient in general. We also characterize the form of the inefficiency: at equilibrium each agent’s policy function weights too heavily the agent’s own preference shock and previous action and not enough the other agents’. The comparative dynamics exercises illustrate e.g., the equilibrium effects of the strength of social interactions and of the social and informational structure of the economy.
Finally, we exploit our characterization results of the equilibria to address generally the issue of identification of social interactions in our context, with population data. While the empirical literature has often interpreted a significant high correlation of socioeconomic choices across agents, e.g., peers, as evidence of social interactions, in the form e.g., of preferences for conformity, it is well known at least since the work of Manski (1993) that the empirical study of social interactions is plagued by subtle identification problems. Intuitively, in our economy for instance, the spatial correlation of actions at equilibrium can be due to social interactions or to the spatial correlation of preference shocks. More formally, take two agents, e.g., agent a and agent b. A positive correlation between xa
t and xbt could be due to e.g., preference for conformity. But the
positive correlation between xa
t and xbt could also be due to a positive correlation between θat
and θb
t. In this last case, preferences for conformity and social interactions would play no role in
the correlation of actions at equilibrium. Rather, such correlation would be due to the fact that agents have correlated preferences. Correlated preferences could generally be due to some sort of assortative matching or positive selection, which induce agents with correlated preferences to interact socially.
In the context of our economy, we ask whether the restrictions implied by the dynamic equilib-rium analysis help identify social interactions and distinguish them from correlated preferences. We show that the answer is in fact affirmative, but only if the economy is non-stationary, in a precise sense. To illustrate our results, consider for instance the issue of peer effects in adolescents’ substance use. Suppose the econometrician observes the behavior of a population of students in a school over time (at different grades). A significant high correlation of socioeconomic choices across students in the school could be due to selection in the endogenous composition of the school in terms of unobserved (to the econometrician) correlated characteristics of the agents. Any significant variation in students’ behavior through time (grades) must however be due to social interactions. A student whose choice is affected by the choices of his school peers will in fact
rationally anticipate how much longer he will interact with them. In particular, his propensity to conform to his peers’ actions will tend to decrease over time (grades) and will be the lowest in the final years in the school. This non-stationarity of each student’s behavior at equilibrium is the key to the identification of social interaction in our class of economies.11
The simplicity of linear models allows us to extend our analysis in several directions which are important in applications and empirical work. This is the case, for instance of general (including asymmetric) neighborhood network structures for social interactions. But our analysis extends also to general stochastic processes for preference shocks and to the addition of global interactions. One particular form of global interactions occurs when each agent’s preferences depend on an average of actions of all other agents in the population, e.g. Brock and Durlauf (2001a), and Glaeser and Scheinkman (2003). This is the case, for instance, if agents have preferences for social status. More generally, global interactions could capture preferences to adhere to aggregate norms of behavior, such as specific group cultures, or other externalities as well as price effects. Finally, and perhaps most importantly, we extend our analysis to encompass a richer structure of dynamic dependence of agents’ actions at equilibrium. In particular we study an economy in which agents’ past behavior is aggregated through an accumulated stock variable which carries habit persistence, which can be directly applied e.g., to the issue of teenage substance addiction due to peer pressure at school. With respect to the addiction literature, as e.g., Becker and Murphy (1988), we model the dynamics of addiction considering peer effects not only in a single-person decision problem, but rather as an equilibrium effect allowing for the intertemporal feedback channel between agents across social space and through time.12 In this context we show that in
equilibrium each agent’s choice depends on the stock of his neighbors’ actions, on their long-term behavioral patterns rather than just on their previous period actions. Also, in non-stationary economies, as the final period approaches, each agent assigns higher weights to his own stock, giving rise to an initiation-addiction behavioral pattern at equilibrium which is consistent with observation, e.g., in Cutler and Glaeser (2007) and DeCicca, Kenkel, and Mathios (2008).
2
Dynamic economies with social interactions
While we develop most of our analysis in the context of linear models, it is useful to set up the general model first, as we do in this section, to be as clear and specific as possible regarding the assumptions we impose on the economy we study.
Time is discrete and is denoted by t = 1, . . . , T . We allow both for infinite economies (T = ∞) and economies with an end period (T < ∞). A typical economy is populated by a countable
11This pattern of behavior appears consistent with the peer effects study of Hoxby (2000a,b). 12
See also Becker, Grossman, and Murphy (1994), Boyer (1978, 1983), Gul and Pesendorfer (2007), Gruber and Koszegi (2001), Iannacone (1986); see Rozen (2010) for theoretical foundations for intrinsic linear habit formation; see also Elster (1999) and Elster and Skog (1999) for surveys.
number of agents a ∈ A.13 Each agent lives for the duration of the economy. At the beginning of
each period t, agent a’s random preference type θa
t is drawn from Θ, a compact subset of a finite
dimensional Euclidean space Rn. The random variables θa
t are independently and identically
distributed across time and agents with probability law ν. We assume, with no loss of generality, that the random variable θt := (θta)a∈A is defined, for all t, on the canonical probability space
(Θ, F, P), where Θ := {(θa)
a∈A : θa ∈ Θ}. At each period t, agent a ∈ A chooses an action xat
from the set X, a compact subset of a finite dimensional Euclidean space Rp. Let X := {x =
(xa)
a∈A: xa∈ X} be the space of individual action profiles.
Each agent a ∈ A interacts with agents in the set N(a), a nonempty subset of the set of agents A, which abstractly represents agent a’s social reference group. The map A : N → 2A is
referred to as a neighbourhood correspondence and is assumed exogenous. Agent a’s instantaneous preferences depend on the current choices of agents in his reference group, {xb
t}b∈N (a),
represent-ing social interactions as direct preference externalities. Agent a’s instantaneous preferences also depend on the agent’s own previous choice, xa
t−1, representing inherent costs to dynamic
be-havioural changes due e.g., to habits. In summary, agent a’s instantaneous preferences at time t are represented by a continuous utility function
xa t−1, xat, {xbt}b∈N (a), θat 7→ uxa t−1, xat, {xbt}b∈N (a), θta
Agents discount expected future utilities using the common stationary discount factor β ∈ (0, 1). The economy has an exogenous initial configuration x0 ∈ X. Let xt−1 = (x0, x1, . . . , xt−1)
and θt−1= (θ
1, . . . , θt−1) be the (t − 1)-period choices and type realizations. Before each agent’s
time t choice, xt−1 is observed by all agents and the current value of the random variable θ t
realizes. Agent a ∈ A observes only the part Iaθt := {θtb : b ∈ I(a)}, where I(a) ⊂ A is
his information set. Similarly, let Iaθt−1 = (Iaθ1, . . . , Iaθt−1). We study both economies with
complete information, I(a) = A, and economies with incomplete information, I(a) A. After each agent’s time t choice, xt= (xbt)b∈A∈ X becomes common knowledge and the economy moves
to time t + 1.
A strategy for an agent a is a sequence of measurable functions xa = (xa
t), where for each
t, xa
t : Xt× (ΘI(a))t → X. Agents’ strategies along with the probability law for types induce a
stochastic process over future configuration paths. Each agent a ∈ A’s objective is to choose xa
to maximize E " T X t=1 βt−1uxat−1, xat, {xtb}b∈N (a), θat (x0, θ1) # (1)
given the strategies of other agents and given (x0, Iaθ1) ∈ X × ΘI(a). 13
We study an economy populated by a countably infinite number of agents where A := Z, but our analysis applies to economies with a finite number of agents.
We require that the social and informational structure satisfies the following symmetry re-strictions:14
1. For all a, b ∈ A, N(b) = Rb−aN (a), where Rb−a is the canonical shift operator in the
direction b − a.15
2. For all a, b ∈ A, I(b) = Rb−aI(a).
We restrict our analysis to symmetric Markov perfect equilibria. Agents’ strategies are Marko-vian if after any t − 1-period history (xt−1, θt), they depend only on the previous period
con-figuration xt−1 and the current type realizations θt. Because of symmetry, it is thus enough to
analyze the optimization problem relative to a single reference agent, say agent 0 ∈ A. Thus, we assume that the optimal choice of any economic agent b ∈ A is determined by a continuous choice function g : X × ΘI(0)× {1, . . . , T } → X such that for all t = 1, . . . , T and after any history
(xt−1, θt) ∈ Xt× Θt, his t-th period choice is given by
xbt(g) xt−1, θt= gT −(t−1)(Rbxt−1, RbI0θt)
The value of the optimization problem of agent a is then given by16
VT g (Rax0, RaI0θ1) = max (xa t)Tt=1 E " T X t=1 βt−1uxa t−1, xat, {xbt(g)}b∈N (a), θta #
The value function associated with this dynamic choice problem can be shown to satisfy Bellman’s Principle of Optimality by standard arguments (see e.g., Stokey and Lucas (1989) ). It can be written in the following recursive form,
VT −(t−1) g (Raxt−1, RaI0θt) (2) = max xa t∈X E " uxa t−1, xat, {xbt(g)}b∈N (a), θat + β VT −t g Ra xa t, n xb t(g) o b6=a , RaI 0θt+1 #
for t = 1, . . . , T and for all (xt−1, θt) ∈ Xt× Θt.17 We are now ready to define our equilibrium
concept.
14Heterogeneity can be incorporated into the probabilistic structure of the types θa
t. Also, we can allow for
heterogeneity of the network structure across agents by augmenting the strategy spaces to incorporate network structure into individual heterogeneity. We explain how we do this in Section 7.1.
15
That is, c ∈ N (a) if and only if c + (b − a) ∈ N (b). Of course, we let A be a linear space when we study symmetric interactions, typically A := Zdthe d-dimensional integer lattice.
16The preference shocks being serially uncorrelated, we do not need to condition on the value of past realizations.
See Section 7.2 for a treatment of persistent shocks.
17
Definition 1 A symmetric Markov Perfect Equilibrium (MPE) of a dynamic economy with social interactions is a measurable mapg∗ : X × ΘI(0)× {1, . . . , T } → X such that for all a ∈ A, for all
t = 1, . . . , T , and for all (xt−1, θt) ∈ Xt× Θt
gT −(t−1)∗ (Rax t−1, RaI0θt) ∈ (3) arg max xa t∈X E " u xat−1, xat,nxbt(g∗)o b∈N (a), θ a t + β VgT −t∗ Ra xat,nxbt(g∗)o b6=a , RaI0θt+1 #
Clearly, an MPE is necessarily a subgame perfect equilibrium; that is, each agent’s continuation strategy is a best response to other agent’s continuation strategies after any possible history. Notice also the time notation we use for the Markovian policy: g∗T −(t−1) denotes the first-period equilibrium choice in a T −(t−1)-periods economy. Since economies are nested, g∗
T −(t−1)represents
also the t-period equilibrium choice in a T -periods economy.
We conclude this section with a few remarks to justify our focus on MPEs. First of all, Markovian strategies are not a restriction for finite-horizon economies: we prove that the unique symmetric subgame perfect equilibrium for any finite-horizon economy is necessarily Markovian. Moreover, in an infinite horizon economy ( T = ∞), a symmetric MPE is not necessarily sta-tionary. The sequence of unique MPEs for finite horizon economies converges however to a g∗ : X × ΘI(0) → X which turns out to be a stationary MPE of the infinite-horizon economy
whose properties we focus on. Finally, we refer to Bisin, Horst and ¨Ozg¨ur (2006) for a discussion of non-Markovian equilibria in a related context.
3
Dynamic Linear Economies with Social interactions and
Con-formity Preferences
We focus our attention on linear economies with conformity preferences. These are environments in which each agent’s preferences incorporate the desire to conform to the choices of agents in his reference group.18
Preferences for conformity arguably provide a rationale for several important social phe-nomena. The empirical literature has for instance documented preferences for conformity as a motivation for smoking and other risky behaviour in teens. Similarly, the role of conformity is also documented by Glaeser, Sacerdote, and Scheinkman (1996) with regards to criminal activity and by a large literature with regards to peer effects in education outcomes.19 Conformity also
represents a natural environment in which to study dynamic equilibrium. In many relevant social
18While we model preferences for conformity directly as a preference externality, we intend this as a reduced
form of models of behavior in groups which induce indirect preferences for conformity, as e.g., Jones (1984), Cole, Mailath and Postlewaite (1992), Bernheim (1994), Peski (2007).
phenomena, in fact, the effects of preferences for conformity are amplified by the presence of lim-its to the reversibility of dynamic choices. This is of course the case for smoking, alcohol abuse and other risky teen behaviour, which are hard to reverse because they might lead to chemical addictions. In other instances, while addiction per se is not at issue, nonetheless behavioural choices are hardly freely reversible because of various social and economic constraints, as is the case, for instance, of engaging in criminal activity. Finally, exogenous and predictable changes in the composition of groups, as e.g., in the case of school peers at the end of a school cycle, introduce important non-stationarities in the agents’ choice. These non-stationarities also call for a formal analysis of dynamic social interactions.
With the objective of providing a clean and simple analysis of dynamic social interactions in a conformity economy, we impose strong(er than required) but natural assumptions.20 In particular
(i ) we restrict the neighborhood correspondence to represent the minimal interaction structure allowing for overlapping groups, (ii ) we restrict preferences to be quadratic, and (iii ) we impose enough regularity conditions on the agents’ choice problem to render it convex. Formally, Assumption 1 A linear conformity economy satisfies the following.
1. Let A := Z represent a general social space. Each agent interacts with his immediate neighbors, i.e., for all a ∈ A, N(a) := {a − 1, a + 1}.
2. The contemporaneous preferences of an agent a ∈ A are represented by the utility function u(xat−1, xat, xta−1, xa+1t , θta) := −α1(xat−1− xat)2− α2(θta− xat)2
−α3(xa−1t − xat)2− α3(xa+1t − xat)2 (4)
where α1, α2, and α3, are positive constants.
3. Let X = Θ = [x, ¯x] ⊂ R, where x < ¯x. Let v be absolutely continuous with a positive density21, E [θa t] = R θa tdν =: ¯θ ∈ (x, ¯x), and V ar(θat) = R θa t − ¯θ 2 dν < ∞.22
Assumption 1-1 requires that the reference group of each agent a ∈ A be composed of his im-mediate neighbors in the social space, namely the agents a − 1 and a + 1. The utility function u defined in Assumption 1-2 describes the trade-off that agent a ∈ A faces between matching his individual characteristics (xa
t−1, θat) and the utility he receives from conforming to the current
choices of his peers (xa−1t , xa+1t ). The different values of αi represent different levels of intensity
of the social interaction motive relative to the own (or intrinsic) motive. Finally, Assumption 1-2 and 1-3 jointly guarantee that the agents’ choice problem is convex.
20See Section 7 for possible directions in which the structure and the results we obtain are easily generalized. 21
We will call a measure µ ‘absolutely continuous’ if it is absolutely continuous with respect to the Lebesgue measure λ, i.e., if µ(A) = 0 for every measurable set A for which λ(A) = 0.
22
We need absolute continuity only when we prove inefficiency. All other results are obtained without that assumption.
3.1 Equilibrium
We provide here the basic theoretical results regarding our dynamic linear social interaction economy with conformity. The reader only interested in the characterization can skip this section, keeping in mind that equilibria exist (for finite economies they are unique) and they induce an ergodic stochastic process over paths of action profiles. Furthermore, a stationary ergodic distribution also exists for the economy. Finally, a recursive algorithm to compute equilibria is derived. The proofs of all statements can be found in the Appendices.
Theorem 1 (Existence - Complete Information) Consider an economy with conformity pref-erences and complete information.
1. If the time horizon is finite (T < ∞), then the economy admits a unique symmetric MPE g∗ : X × Θ × {1, · · · , T } 7→ X such that for all t ∈ {1, . . . , T }, for all (xt−1, θt) ∈ X × Θ
g∗T −(t−1)(xt−1, θt) = X a∈A caT −(t−1)xat−1+X a∈A daT −(t−1)θat + eT −(t−1)θ P − a.s. where ca
τ, daτ, eτ ≥ 0, a ∈ A, and eτ +Pa∈A(caτ + daτ) = 1, 0 ≤ τ ≤ T . Moreover, the
equilibrium is also unique in the class of subgame perfect equilibria (SPE), meaning that there does not exist any non-Markovian SPE for our economy.
2. If the time horizon is infinite (T = ∞), then the economy admits a symmetric stationary MPEg∗: X × Θ 7→ X such that
g∗(xt−1, θt) = X a∈A caxat−1+X a∈A daθat + e θ P − a.s.
where ca, da, e ≥ 0, for a ∈ A, and e +P
a∈A(ca+ da) = 1.23
The theorems in this section can be extended with straightforward modifications to the case of incomplete information. We state without proof, e.g., the existence theorem for economies with incomplete information next.
Theorem 2 (Existence - Incomplete Information) Consider an economy with conformity preferences and with incomplete information.
1. ForT < ∞, the economy admits a unique symmetric MPE g∗ : X×ΘI(0)× {1, · · · , T } 7→ X
such that for all t ∈ {1, . . . , T }, g∗T −(t−1)(xt−1, I0θt) = X a∈A caT −(t−1)xat−1+ X a∈I(0) daT −(t−1)θta+ eT −(t−1)θ P − a.s. where ca τ, daτ, eτ ≥ 0 and eτ+Pa∈Acaτ+ P a∈I(0)daτ = 1, 0 ≤ τ ≤ T . 23
Several assumptions can be relaxed while guaranteeing existence. In particular, the symmetry of the neighbor-hood structure can be substantially relaxed. See Section 7.1 for the discussion.
2. For T = ∞, the economy admits a symmetric MPE g∗: X × ΘI(0)7→ X such that g∗(xt−1, I0θt) = X a∈A caxat−1+ X a∈I(0) daθta+ e θ P − a.s. where ca, da, e ≥ 0 and e +P a∈ Aca+ P a∈I(0)da= 1.
3.1.1 A Sketch of the Proof
The proof of the existence theorem requires some subtle arguments. While referring to the Appendix for details, a few comments here in this respect will be useful. Consider the (infinite dimensional) choice problem of each agent a ∈ A. To be able to apply standard variational arguments to this problem it is necessary to bound the marginal effect of any infinitesimal change dxa on the agent’s value function. To this end, the Envelope theorem (as e.g., in Benveniste and
Scheinkman (1979)) is not enough, as dxa affects agent a’s value function directly and indirectly,
through its effects on all agents b ∈ A\a’s choices, which in turn affect agent a’s value function. The marginal effect of any infinitesimal change dxa is then an infinite sum. Furthermore, each
term in the sum contains endogenous terms from some agent b ∈ A\a’s policy function (and there is an infinite number of them), which makes it impossible to adopt the methodology used by Santos (1991) to prove the smoothness of the policy function in infinite dimensional recursive choice problems. In our economy, with quadratic utility, policy functions are necessarily linear and, provided we show that equilibria are interior, symmetric MPE’s can be represented by a policy function which is obtained as a fixed point of a recursive map which can be directly studied. Extending the existence proof to general preferences would require therefore sufficient conditions on the structural parameters to control the curvature of the policy function of each agent’s decision problem. We conjecture that this can be done although sufficient conditions do not appear transparently from our proof. A more detailed sketch of the steps involved in the existence proof follows.
Step 1 In the last period (1-period continuation) of any finite-horizon economy, first order condi-tions (FOC’s) induce a contraction operator on the space of bounded measurable funccondi-tions having as arguments any t-length history. Hence, there exists a unique symmetric (possibly history-dependent) equilibrium. We then show that the equilibrium policy must be Marko-vian and should take the convex combination form in the statement of Theorem 1 (Lemma 1).
Step 2 For any finite horizon (T < ∞) economy, we assume that in the continuation from period 2 on agents choose according to the unique symmetric MPE, g : X × Θ × {1, · · · , T − 1} 7→ X. Linearity of the policy in the continuation keeps a generic agent’s dynamic program strictly concave and FOC’s are necessary and sufficient for a pure strategy maximum. We show
that we can write FOC’s as functions only of first period choices and preference and the mean preference shocks. By the same token as in Step 1, we focus on Markovian strategies. FOC induces a contraction operator on the set of Markovian strategies into itself. Hence, there exists a unique fixed point gT∗ : X × Θ 7→ X. We conclude that for the T -period economy, the map (g∗
T, g) : X × Θ × {1, 2, · · · , T } 7→ X is the unique symmetric MPE in
pure strategies and has the convex combination form as in the statement of the theorem, which completes the induction argument.
Step 3 The final step involves taking a limit. We construct a series of finite economies, approximat-ing the ∞-horizon economy, given an appropriate topology. We then show that, the finite truncation equilibrium correspondence is upper-hemi-continuous (u.h.c.) with respect to the parametrization. This is however not enough for stationarity. We prove that the behav-ioral Markovian strategy set (the set G) is compact. This helps us prove that the sequence of finite-horizon equilibrium policy functions converges uniformly to a policy function in G (which is an equilibrium policy due to u.h.c of the equilibrium correspondence), hence the same one every period, after any history. This gives us stationarity.
3.2 The parameters of the policy function
By exploiting the linearity of policy functions, our method of proof is constructive, producing a direct and useful recursive computational characterization for the parameters of the symmetric policy function at equilibrium. We repeatedly exploit this characterization in the next section e.g., when performing comparative dynamics exercises. Consider the choice problem of agent 0. For any T -period economy, agent 0’s dynamic program yields a FOC that takes the following form (see Lemma 3)
x01 = ∆−1T α1x00+ α2θ01+ X a6=0 γTb xb1+ µTθ¯ (5)
where ∆T and γTb, and µT are the effects on agent zero’s discounted expected marginal utility of
changes in agents 0 and b’s first period actions and the change in the level of ¯θ, respectively. Let Lc,d,e:= {(c, d, e) : e ≥ 0, ca≥ 0, da≥ 0, ∀a and e +
X
a
(ca+ da) = 1}
be the space of nonnegative coefficient sequences whose sum is 1. The existence of an equilibrium policy for the first period of a T -period economy is then equivalent to the existence of a coefficient sequence (c∗
(ˆc, ˆd, ˆe) = LT(c, d, e) and for each a ∈ A ˆ ca = ∆−1 T α11{a=0}+Pb6=0γTb ca−b ˆ da = ∆−1 T α21{a=0}+ P b6=0γTb da−b ˆ e = ∆−1T µT + ePb6=0γTb (6)
by matching coefficients of the policy on both sides of (5). The parameters of the map LT,
namely ∆T, γTb
a6=0, µT, depend only on the continuation equilibrium coefficients (c ∗
s, d∗s, e∗s)T −1s=1
in a linear fashion (see (31), (48), and (50) for their detailed expressions). For T = 1, the parameters of L1are dictated directly by the underlying preferences, namely ∆1 = α1+ α2+ 2α3,
γ1 1 = γ
−1
1 = α3, γ1b = 0, for all b 6= −1, 0, 1, and µ1 = 0. Thus, the map L1 defined by the system
in (6) becomes
ˆ
ca = ∆−1
1 α11{a=0}+ α3ca−1+ α3ca+1
ˆ
da = ∆−1
1 α21{a=0}+ α3da−1+ α3da+1
ˆ
e = ∆−11 (2α3e)
(7)
which is a contraction mapping whose unique fixed point is computed as the unique root to a second-order difference equation that satisfies transversality conditions toward both infinities. Consequently, the equilibrium policy coefficients are computed as in the next Theorem.
Theorem 3 (Recursive algorithm) Consider a finite-horizonT -period economy with confor-mity preferences (αi > 0, i = 1, 2, 3) and complete information.
(i) The map L1 for a one-period economy, defined in (6), forms a second-order difference
equation for the equilibrium coefficient sequence, whose unique non-explosive, exponential solution is the unique fixed point ofL1. We compute the coefficient sequence in closed-form.
For anya ∈ A, c∗a1 = r1|a| α1 α1+ α2 1 − r1 1 + r1
and d∗a1 = r1|a| α2 α1+ α2 1 − r1 1 + r1 (8) where r1= ∆1 2 α3 − s ∆1 2 α3 2 − 1 with∆1= α1+ α2+ 2 α3.
(ii) The coefficients (c∗s, d∗s, e∗s)Ts=2 of the sequence of Markov equilibrium polices are computed recursively as the unique fixed points of the recursive contraction mapsLs : Lc,d,e→ Lc,d,e,
s = 2, . . . , T , defined in (6), whose parameters ∆s, (γsa)a6=0, µs depend linearly only on the
continuation equilibrium policy coefficients(c∗
(iii) Moreover, limT →∞(c∗T, d∗T, e∗T) = (c∗, d∗, e∗) exists and it is the coefficient sequence of the
stationary Markovian equilibrium policy function for the infinite-horizon economy whose existence is proved in Theorem 1.
Fixed point calculations take less than a few seconds on an ordinary computer, for each period. Finally, the sequence of fixed point maps that we compute at each iteration converges to a policy sequence, which turns out to be the infinite-horizon stationary MPE. The convergence is very rapid, under a few minutes.
0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 Agent a Co e ffi c ie n ts (c a T)i n th e o p ti m a l p o li cy T=1T=2 T=3 T=4 T=5 T='
Figure 1: Non-stationary Optimal Policy.
3.3 Ergodicity
With such characterization of the parameters of the policy function at hand, we are able to char-acterize very tightly the spatial (cross-sectional) and intertemporal behavior of the equilibrium process emerging from the class of dynamic models we study. Let π0 be an initial distribution on
the configuration space X. Given the initial distribution π0, a stationary MPE of the economy
with conformity induces an equilibrium process (xt∈ X)∞t=0 (via the policy function g∗) and an
associated transition function Qg∗. This latter generates iteratively a sequence of distributions
(πt)∞t=1 on the configuration space X, i.e., for t = 0, 1, . . .
πt+1(A) = πtQg∗(A) =
Z
X
Qg∗(xt, A) πt(dxt+1) (9)
We show first that, given the induced equilibrium process, the transition function Qg∗ admits an
π is ergodic24.
Ergodicity does not necessarily imply the convergence of the equilibrium process to a unique distribution starting from an arbitrary initial distribution π0. Conditions are necessary to
guar-antee such convergence.25 We next show that, for any initial distribution π
0 and a stationary
Markovian policy function g∗, the equilibrium process (x
t∈ X)∞t=0converges in distribution to the
invariant distribution π, independently of π0.26 This also implies that π is the unique invariant
distribution of the equilibrium process (xt∈ X)∞t=0. More specifically,
Theorem 4 (Ergodicity) Suppose the process (θa t)∞t=−∞
a∈A is i.i.d. with respect to a and t
according to ν. The equilibrium process (xt∈ X)∞t=0 induced by a symmetric stationary Markov
perfect equilibrium of an economy with conformity via the policy function g∗(xt−1, θt) and the
unique invariant measure π as the initial distribution is ergodic; π is the joint distribution of
xt= e θ 1 − C + ∞ X s=1 X b1∈A · · · X bs∈A cb1· · · cbs−1dbsθa+b1+···+bs t+1−s a∈A (10)
where C :=Pa∈Aca is the sum of coefficients in the stationary policy function that multiply
cor-responding agents’ last period choices. Moreover, the sequence(πt)∞t=1 of distributions generated
by the equilibrium process (xt ∈ X)∞t=0 converges to π in the topology of weak convergence for
probability measures, independently of any arbitrary initial distributionπ0.27
4
Characterization of equilibrium
Exploiting the linear structure of our economies we can study equilibria in some detail. Recall that the policy function in each period t = 1, . . . , T , for each agent a ∈ A, is
xat =X b∈A cbT −(t−1)xa+bt−1 +X b∈A dbT −(t−1)θa+bt + eT −(t−1)θ, (11) with eT −(t−1)+ P a∈A(caT −(t−1)+ d a
T −(t−1)) = 1, when T is finite; and 24We call a Markov process (x
t) with state space X under a probability measure P ergodic if T1 PTt=1f (xt) →
R f dP P -almost surely for every bounded measurable function f : X → R. See for example Blume (1982), Duffie et al (1994) and Hansen (1982) for the use of ergodicity in dynamic economic theory and modern econometric theory.
25
The well-known D¨oblin conditions to that effect can be found in Doob (1953). See also Futia (1982), Neveu (1965), and Tweedie (1975) for similar characterizations.
26Note however that Theorem 1 does not guarantee that the policy function g∗
(xt−1, θt) is unique. 27
A sequence of probability measures (λt) is said to converge weakly (or in the topology of weak convergence for
probability measures) to λ if, for any bounded, measurable, continuous function f : X → R, limt→∞R f dλt=R f dλ
xat =X
b∈A
cbxa+bt−1 +X
b∈A
dbθa+bt + e θ, (12)
with e +Pa∈A(ca+ da) = 1, in the infinite-horizon case.
First of all, we study the parameters of the policy function. The coefficients cb
T −(t−1) and
db
T −(t−1) (resp. c
b and db in the case of infinite-horizon economies), in particular, may be viewed
as a measure for the total impact of the action xa+bt−1 and of the preference shock θa+bt of agent a+b, respectively, on the optimal current choice of agent a; where b concisely represents the social distance between the two agents.28 Furthermore, we study a fundamental statistical property
of equilibrium, cross-sectional auto-correlation of actions. In fact, although any agent a ∈ A interacts directly only with a small subset of the population, at equilibrium, each agent’s optimal choice is correlated with those of all the other agents. Let ρa,T denote the conditional correlation
between the first-period equilibrium actions of agents a-step away from each other, in the T -period economy, given x0 ∈ X:29 ρa,T = Covx0 1, xa1 x0 V arx0 1 x0 . (13) 4.1 Policy Function
Consider first a finite-horizon economy. Since the policy function for this economy is well-defined, the coefficients cb T −(t−1) and d b T −(t−1) satisfy lim |b|→∞c a+b T −(t−1)= lim|b|→∞d a+b T −(t−1)= 0
The impact of an agent a + b on agent a tends to zero as |b| → ∞. In this sense, linear conformity economies display weak social interactions.
Furthermore, as we have shown in Section 3.2, lim
T →∞ cT = c, limT →∞ dT = d, and limT →∞eT = e 28See Akerlof (1997) for richer definitions of social distance.
29
The correlation between the first-period optimal choices of agents a and b, is Covxa1, x b 1 x0 r V ar xa 1 x0 V ar xb 1 x0 .
Due to the symmetry imposed on our economy, such correlations are independent of agents’ labels but depends only on |b − a|. Consequently, we can define the conditional correlation function with distances computed relative to any agent, in particular agent 0.
The finite-horizon parameters converge (uniformly) to the infinite-horizon stationary policy pa-rameters.
Finally, equilibrium policy functions are non-stationary in the finite economy, as rational forward-looking agents change their behavior optimally through time. In the final periods, for example, social interactions lose weight relative to individual characteristics; see Figure 1.30
4.2 Cross-sectional Auto-correlations
Exploiting the equilibrium characterization provided by Theorems 1 and 3, and the independence of preference shocks across agents, we can compute the covariance terms:
Covx01, xa1 x0 = V ar(θ) X a1∈A da1 T d a1−a T . (14) The expressionPa 1∈Ad a1 T d a1−a
T is the discrete self-convolution of the equilibrium policy sequence
dT = daT1
a1∈A, where a acts as the shift parameter.
31 In Figure 2 we show how the convolution
behaves with respect to the distance a, for the same set of parameters as in Figure 1. Substituting the form in (14) back in (13) for both terms, we obtain
0 a 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Agents Co e ffi c ie n ts (d a T)i n th e o p ti m a l p o li cy d0 T d ï a T dT ï 1 dT aï1
Figure 2: Convolution of the Policy Coefficient Sequence.
30We plot in Figure 1 only one side of the policy coefficient sequence to get a close-up view of the change in
equilibrium behavior. The left hand side is the mirror image of that due to symmetry. Parameter values for this figure are α1
α3 = 1, α3
α2 = 10, and β = .95 31
ρa,T = P b∈AdTbdTb−a P b∈AdTb dTb (15)
the a-step conditional cross-sectional autocorrelations for the first-period equilibrium choices of the T -period economy. Exploiting the recursive algorithm provided by Theorem 3, we can com-pute these autocorrelations easily for any finite economy. We can then study the behavior of the conditional correlation function ρa,T through time (T ) and across social space (a). These
corre-lations exhibit interesting dynamics: they are declining in a, for any T , but the rate of decline cannot be ranked in T , given a; see Figure 3 for an example with the same parametrization we used above for the policy weights in Figure 1. In particular, given a T -period economy, consider
0 1 2 3 4 5 6 7 8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Distance b etween agents, a
Au to-cor re lat ion ρa, T T=1 T=2 T=3 T=4 T=5 T='
Figure 3: Cross-sectional Auto-correlations. the T -period rate of convergence of the spatial autocorrelations, for a ≥ 032
ra,T =
ρa+1,T
ρa,T
.
It is easy to show analytically that ra,1 declines monotonically and becomes constant at the tail
in a.33 On the other hand, r
a,T is typically non-monotonic in a, for longer horizons, including for
T = ∞; see Figure 4.
Finally, consider the T -period rate of tail convergence of the spatial autocorrelations, rT := lim
a→∞ra,T = lima→∞
ρa+1,T
ρa,T
Similarly, let the same rate for the infinite-horizon economy (T = ∞) be represented by r.
32
The rate is symmetrically defined with respect to agent 0, i.e., ra,T =
ρa−1,T
ρa,T , for any a ≤ 0. 33
5 10 15 20 25 0.4 0.45 0.5 0.55 0.6 0.65 0.7
Distance b etween agents, a
Ra ti o ! ρ a +1 ,T ρa, T " T=1 T=2 T=3 T=4 T=5 T='
Figure 4: Rate of Convergence of the Auto-correlations.
Proposition 1 (Tail Convergence Monotonicity) 34 The rate rT is monotone increasing
with respect to the length of the economy,
rT +1 > rT, for finite T ≥ 1.
Moreover, the sequence of tail convergence rate for finite-horizon economies converges to that of the infinite-horizon economy as the horizon length gets larger and the limit rate is strictly less than 1:
lim
T →∞rT = r < 1.
In other words, even though the autocorrelation functions might behave non-monotonically for shorter social distances, they eventually converge (as social distance a → ∞) to an exponential rate in the tail. Moreover, rates of tail convergence are higher the farther is the final period of the economy (as T → ∞). This is because rational agents choose to correlate their actions more with their neighbors in early periods and progressively less so as they approach the end of their social interactions. Finally, as the infinite-horizon limit is approached, the rate of tail convergence becomes stationary (as to be expected since finite-horizon equilibria approximate the stationary infinite-horizon equilibrium). We use this intuition to the fullest extent when discussing identification in Section 6.
In an infinite-horizon economy social interactions manifest themselves at the stationary er-godic distribution by means of spatial autocorrelation of actions. Given x0 ∈ X, the conditional
covariance in period t of an infinite-horizon economy, between two agents a agents away from each
34
other is denoted by Covx0 t, xat x0
. Let Cov x0, xa be the a-step unconditional covariance
at the ergodic stationary distribution. Since the stationary MPE is ergodic, it is easy to see from Lemma 2 (i) and Theorem 4 that as t gets arbitrarily large, the conditional t-period covariance between agents 0 and a converges to its unconditional counterpart at the limit distribution, i.e.,
Cov x0, xa= lim
t→∞Cov
x0t, xat x0
Moreover, the limit unconditional correlation ρb between the actions of agents a and a + b is
independent of x0 and it satisfies
ρb = Cov x0, xa V ar (x0) = limt→∞ Covx0 t, xat x0 V arx0 t x0
Finally, because of the stationarity of the policy function in (12), the limit covariance between two agents a agents away from each other can be written as
Cov x0, xa = lim t→∞Cov x0 t, xat x0 = X a1∈A X b1∈A ca1cb1Cov xa1, xa+b1 + V ar(θ) X a1∈A da1da1−a, (16)
and hence it has a simple recursive structure. In fact, since the sum of the stationary weights multiplying covariances on the right hand side are strictly less than one, this system can be seen as a contraction operator. Hence, for each one-step conditional autocorrelation sequence, there is a unique stationary unconditional autocorrelation sequence that we can compute using the above recursive system easily. We later exploit this recursive structure further in Section 6.1 when we compare equilibrium stationary distributions induced by myopic and rational agents.
In Figure 5, we report the correlation functions in both the mild and strong conformity parameterizations as a function of social distance, b.35 Two effects are worth mentioning here.
Firstly, both correlation functions converge to zero as the distance between two agents become arbitrarily large. Secondly, this convergence is much faster in the case of mild interactions than in the case of strong interactions. For example, the correlation between the equilibrium choices of agent a and agent a + 3 (or a − 3 due to symmetry) is about 7% in the case of mild interactions whereas it is about 75% in the case of strong interactions. The correlation between the equilibrium choices of agent a and agent a + 6 are about 0% and 40% respectively. The strength of the desire to conform built in individuals’ preferences determine endogenously, at equilibrium, the size of the effective neighborhood with which an individual interacts.
35
0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Distance between two agents, b
Correlation Coefficient,
ρb
Mild Strong
Figure 5: Correlation function at the ergodic distribution for Mild and Strong Interactions.
5
Equilibrium Properties and Comparative dynamics
In this section we first study the welfare properties of equilibrium and then we use the character-ization of equilibria we obtained to produce several simulations illustrating various comparative dynamics exercises of interest.
5.1 (In)efficiency
Social interactions are modelled in this paper as a preference externality, that is, by introducing a dependence of agent a’s preferences on his/her peers’ actions. Not surprisingly, therefore, equi-libria will not be efficient in general. In this section we also characterize the form the inefficiency takes when social interactions are modelled as preferences for conformity.
A benevolent social planner, taking into account the preference externalities and at the same time treating all agents symmetrically, would maximize the expected discounted utility of a generic agent, say of agent a ∈ A, by choosing a symmetric choice function h ∈ CB(X × Θ, X), the space of bounded, continuous, and X-valued measurable functions. In other words, h is the solution
to36 max {f ∈CB(X×Θ,X)} Z "XT t=1 βt−1 − α1 fT −(t−2)(Raxt−2, RaI0θt−1) − fT −(t−1)(Raxt−1, RaI0θt) 2 −α2 θta− fT −(t−1)(Raxt−1, RaI0θt) 2 −α3 fT −(t−1)(Ra−1xt−1, Ra−1I0θt) − fT −(t−1)(Raxt−1, RaI0θt) 2 −α3 fT −(t−1)(Ra+1xt−1, Ra+1I0θt) − fT −(t−1)(Raxt−1, RaI0θt) 2 !# T Y t=1 P (dθt) π0(dx0)
where π0 is an absolutely continuous distribution on the initial choice profiles with a positive
density. This problem can be written recursively. For any agent a ∈ A, for all t = 1, . . . , T , and all (xT −1, θT) ∈ X × ΘI(0), let the value of using the choice rule h in the continuation be defined
as Vh,T −(t−1)(Rax T −1, RaθT) = −α1 x0t−1− hT −(t−1)(Raxt−1, RaI0θt) 2 −α2 θat − hT −(t−1)(Raxt−1, RaI0θt) 2 −α3 hT −(t−1)(Ra−1xt−1, Ra−1I0θt) − hT −(t−1)(Raxt−1, RaI0θt) 2 −α3 hT −(t−1)(Ra+1xt−1, Ra+1I0θt) − hT −(t−1)(Raxt−1, RaI0θt) 2 +β Z Vh,T −tRanht(Rbxt−1, RbI0θt) o b∈A, R aI 0θt+1 P (dθt+1)
which leads us to the following definition
Definition 2 (Recursive Planning Problem) Let a T -period linear economy with social in-teractions and conformity preferences be given. Let π0 be an absolutely continuous distribution
on the initial choice profiles with a positive density. A symmetric Markovian choice function g : X × ΘI(0)× {1, . . . , T } → X is said to be efficient if it is a solution, for all a ∈ A, and for
allt = 1, . . . , T , to arg max {f ∈CB(X×Θ,X)} Z " − α1 x0t−1− hT −(t−1)(Raxt−1, RaI0θt) 2 −α2 θat − hT −(t−1)(Raxt−1, RaI0θt) 2 −α3 hT −(t−1)(Ra−1xt−1, Ra−1I0θt) − hT −(t−1)(Raxt−1, RaI0θt) 2 −α3 hT −(t−1)(Ra+1xt−1, Ra+1I0θt) − hT −(t−1)(Raxt−1, RaI0θt) 2 +β Vh,T −tRa nh(Rbxt−1, RbI0θt) o b∈A, R aI 0θt+1 # P (dθt) P (dθt+1) πt(dxt−1)
where πt is the distribution on t-th period choice profiles induced by π0 and the planner’s choice
rule h.
36
As noted, preferences for conformity introduce an externality in each agent a ∈ A’s decision problem, which depends directly on the actions of agents in neighbourhood N (a) and, indirectly, on the actions of all agents in the economy. In equilibrium, agents do not internalize the impact of their choices on other agents today and in the future. More precisely,
Theorem 5 (Inefficiency of equilibrium) A symmetric MPE of a conformity economy is in-efficient.
Furthermore, an efficient policy function will tend to weight less heavily the agent’s own-effect and more heavily other agents’ effects, relative to the equilibrium policy. This effect, hence the inefficiency, are neatly exhibited by comparing the equations determining policy weights in the planner (36) and equilibrium (24) scenarios. The (absolute value of the) weights the planner’s equation associates on neighbors’ choices is twice as large as the weights associated to neighbors in the equilibrium equation ( α3
α1+α2+2α3
). As a consequence, the relative weights that the planner assigns to neighbors’ choices are always higher than the ones that each agent uses in equilibrium.37
aï5 aï1 a a+1 a+5
0.05 0.1 0.15 0.2 0.25 AGENT
Weight in the Policy Function
ceqbm
cplanner
Figure 6: Inefficiency of equilibrium.
A graphic representation of the inefficiency is obtained in Figure 6, which presents the coef-ficient plot for the equilibrium policy of a one-period economy (equivalently the final period of any finite-horizon economy): ceqbm (blue dots), and for the planner’s solution, cplanner(red dots),
respectively, for a given agent a ∈ A, and for a given set of parameter values (α1 α2 =
α2
α3 = 1, and 37
Normalizing the relative coefficients to form a probability measure (see the argument in the proof of Lemma 2 (iv)), we have that the measure obtained from the planner’s policy is a mean-preserving spread of the measure obtained at equilibrium.
a-15 a-10 a-5 a a+5 a+10 a+15 0 0.05 0.1 0.15 0.2 0.25 AGENT
Weight in the Policy
strong mild c0 c0 c-5 c 10
Figure 7: Weights on past history in the stationary policy function.
β = .95).38
5.2 Comparative Dynamics: Peer Effects
The strength of the agents’ preferences for conformity depends on the size of the preference parameter α3relatively to α1and α2. A policy function is represented in Figure 7, which compares
a case with mild preferences for conformity (with parametrization α1 α2 =
α2 α3 = 1)
39 with one with
strong preferences for conformity (with parametrization α1 α2 = 1,
α2 α3 =
1
20). On the x-axis,
we plot agent a and his neighbors, while on the y-axis, we plot the weights (cb)
b∈A that the
symmetric policy function g associates with the last period actions of agents (a + b)b∈A. While
each agent’s interaction neighborhood is only composed of two agents, in effect local interactions involve indirectly larger groups. How large are the groups depends endogenously on the strength of the agents’ preferences for conformity. Notice e.g., that in Figure 7, local interactions involve effectively a group of about ten neighbors when preferences for conformity are mild and involve a group of about thirty neighbors when preferences for conformity are strong. Furthermore, for the same cases of mild and strong conformity, we compare in Figure 8 the case in which neighborhoods are overlapping, N (a) = {a − 1, a + 1}, with the case of non-overlapping one-sided neighborhoods, N (a) = {a+1}.40 Two effects are present here. Firstly, as in Figure 7, an increase
38We call this parametrization the mild-interaction case in Section 5.2. 39
The discount rate is fixed at β = .95 in all the simulations unless mentioned otherwise.
40In this case, the policy function is
xat = g(R a xt−1, Raθt) = X b≥0 cbxa+bt−1+ X b≥0 dbθta+b+ e θ.
a-15 a-10 a-5 a a+5 a+10 a+15 0 0.05 0.1 0.15 0.2 0.25
Weight in the policy
(mild)
a-15 a-10 a-5 a a+5 a+10 a+15
0 0.05 0.1 0.15 0.2 0.25 0.3 AGENTS
Weight in the policy
(strong)
two one
two one
Figure 8: One-sided vs. two-sided interactions.
in the strength of the interaction parameter spreads the interaction effects over a larger social geography. Secondly, this spread is observed most significantly in the case of non-overlapping neighborhoods due to the uni-directional nature of the interactions. In turn, spatial correlations induce correlated actions of agents in endogenously formed groups.
At the ergodic stationary distribution, when the dependence of the agents’ actions in equilib-rium are independent of the initial configuration of actions x0, such correlations in endogenously
formed groups is manifested in a phenomenon which we refer to as local norms of behavior (see Figure 9).41 In Figure 9, we plot 100 neighboring agents on the x-axis and their optimal choices
drawn from the limit distribution at the same future date, on the y-axis. In the top panel, clearly the optimal actions are more spread and do not follow a significant pattern. In the bottom panel though, the optimal choices are more concentrated and follow a clear path. This is due to the fact that, in equilibrium agents conform to the actions of neighboring agents, leading the way to the creation of similar local behavior. In the bottom panel of Figure 9, we observe groups of agents (e.g., in the neighborhood of agent 20) choosing relatively low actions, and other groups (e.g., in the neighborhood of agent 70) choosing instead high actions. Two interesting aspects of this phenomenon are firstly that every individual uses the same symmetric policy function to make his choices and all heterogeneity is captured by random types and we still have high spatial
correlation and high spatial variation. Secondly, the initial configuration of actions is irrelevant since the limit distribution of individual actions in this economy is ergodic.
0 10 20 30 40 50 60 70 80 90 100 −1.5 −1 −0.5 0 0.5 1 1.5 0 10 20 30 40 50 60 70 80 90 100 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4
Figure 9: Ergodic Limit of Mild (top) and Strong (bottom) Interactions for 100 adjacent agents.
5.2.1 Comparative Dynamics: Information
In Figure 10, we compare the case in which agents have complete information with the case in which they have incomplete information. In this last case, the policy function is
xat = g(Raxt−1, RaI0θt) = X b∈A cbxa+bt−1 + X b∈I(0) daθa+bt + e θ.
In particular, we record the effect of an expansion of the information set RaI
0θt (individuals
whose types are observed by agent a) on best responses. We start with an information structure in which each agent observes his own type only. We then increase the number of types observed by
each agent a (maintaining the symmetry of two-sided interactions) up to the complete information limit. The red dots represent the optimal weights in the policy function of an of agent a as a
M M M H I H I H I H I M (a) (b) (c) (d)
Figure 10: Effect of Information on Interactions.
response to the informational structure. The lower left vertex represents (H)istory, the total sum of weights assigned to last period’s choices. The lower right vertex represents (I)nformation, the sum of weights on current types observed. Finally, the upper vertex represents average information, (M)ean type, ¯θ. In part (a), we have mild preferences for conformity once again. The dots are concentrated near the middle of the triangle (equal weights on history, information, and mean type) and they do not move much as a response to changes in the amount of current information. Part (d) is the counterpart with strong interactions. Hence the significant change from almost no weight on current information to almost equal weights. Individuals use the information in the best possible way by putting more weight on it in their policy functions. This is due to the fact that forming expectations more precisely how the neighbors will behave becomes more important for each agent, due to the increased strength of interactions. Part (c) is mild interactionsbut strong own-type effect (α1
α2 = 1 20,
α2
α3 = 20) and part (b) is strong interactions and
strong own-type effect(α1 α2 =
1 20,
α2
α3 = 1). We do not see much change in (b), although most of
the total weight is put on information. This is mainly due to the fact that any agent a cares so much about his current type that, he neglects the other effects. In (c), although the own-effect is still strong, due to the strength of interactions, each agent uses the average information to form the best expectations regarding the behavior of the other agents. As the amount of information increases, each agent forms better expectations by transferring the policy weight from average information to precise information on close neighbors.
6
Identification
We study here the identification of social interactions in the context of our linear dynamic economy with conformity. Identification obtains when the restrictions imposed on actions at equilibrium by preferences for conformity are distinct from those imposed by other relevant structural mod-els.42 Consider in particular an alternative structural model characterized by (cross-sectionally)
correlated preferences across agents. This specific alternative model is focal because correlated preferences could be generally due to some sort of assortative matching or positive selection in so-cial interaction, which induces agents with correlated preferences to interact soso-cially. Suppose an econometrician observes panel data of individual actions over time displaying spatial correlation of individual actions at each time. Such correlation can generally be due to social interactions, as our analysis has shown. Such correlation could also ensue, however, from the spatial correlation of preference types, which we have excluded by assumption in our analysis to this point. But is there any structure in the spatial correlation which is implied by preference for conformity and not by correlated preferences? An affirmative answer to this question implies that the social interaction model is identified with respect to the correlated preferences model.
The structural analysis of identification in linear economies with social interactions starts with Manski (1993).43 Manski restricts his analysis to static linear models, or, more specifically,
linear economies in which the social interactions operate through the mean action in a pre-specified group, (linear in means models). In this context, identification is problematic due to the colinearity problem introduced by the mean action, the so-called reflection problem, and due to the possible correlation of unobservables. In the context of linear in means models, a recent literature has studied identification under the condition that the population of agents could be partitioned into a sequence of finitely-populated non-overlapping groups; see e.g., Graham and Hahn (2005).44
The economies we study in this paper are related to those studied by Manski (1993) and others in that we maintain linearity, an assumption which renders identification harder (see Blume et
42The question of identification in economics has been clearly defined by Koopmans (1949) and Koopmans and
Reiersøl (1950). The issue of identification goes back to Pigou (1910), Schultz (1938), Frisch (1928, 1933, 1934, and 1938), and Frisch et al (1931). By identification we mean identification in population (Sometimes identification in population is called identifiability ; see e.g., Chiappori and Ekeland, 2009). See also Marschak (1942), Haavelmo (1944), Koopmans, Rubin, and Leipnik (1950), Wald (1950), Hurwicz (1950). More recent surveys on the topic exist of course; see Rothenberg (1971), Hausman and Taylor (1983), Hsiao (1983), Matzkin (2007), and Dufour and Hsiao (2008).
43
Blume et al. (2011), Blume and Durlauf (2005), Brock and Durlauf (2007), Graham (2011), and Manski (1993, 2000, 2007) survey the main questions pertaining to identification in this social context.
44Also: Davezies, D’Haultfoeuille and Foug`ere (2009) extends these results exploiting variation over the size of
the populations; Graham (2008) uses excess variance across groups; Bramoull´e et al (2009) uses reference group heterogeneity for identification. Other recent contributions include Glaeser and Scheinkman (2001), Graham and Hahn (2005); De Paula (2009), Evans, Oates and Schwab (1992), Ioannides and Zabel (2008), and Zanella (2007).
al. (2011)). On the other hand we introduce several fundamental distinguishing features: in particular, we allow for more general forms of social interactions across agents and for dynamic economies. More precisely, in the class of economies we study, the equilibrium action of agent a in an infinite horizon economy satisfies
xat = β1xat−1+ β2θta+
X
b6=0
β3,bxa+bt ;
while in a linear in means economy the corresponding equation is:45
xa= βθa+ γ X
b∈N (a)
xb.
By studying populations composed of an infinite number of overlapping neighborhoods our analysis sheds some light on the nature of identification results which exploit an infinite number of non-overlapping groups, as in Graham and Hahn (2005) and in the literature discussed in footnote 44. The overlapping structure of our neighborhoods, in fact, breaks the independence which is required when non-overlapping groups are considered.46 Furthermore, by studying
dy-namic models we are able to exploit the theoretical implications deriving from the optimality of the dynamic choices of agents on time series autocorrelations of actions, over and above the implications regarding the cross-sectional (spatial) correlations. In a related context, de Paula (2009) and Brock and Durlauf (2010) also exploit the properties of dynamic equilibrium, the discontinuity in adoption curves in their continuous time model, to identify social interactions.47
We turn to our main identification results. The first series of results regards the identification of the dynamic structure - that is, distinguishing the properties of dynamic social interaction economies from those of myopic (hence static) economies. The second series of results regards instead the identification of social interactions, that is, distinguishing preference for conformity from correlated preferences.
6.1 Dynamic Rationality vs. Myopia
In this section we compare equilibrium configurations of dynamic economies with rational agents with those of economies with myopic agents. When agents are myopic, even economies with a dynamic structure, e.g., when agents’ actions at time t depend on their previous actions, are
45
Note that, to ease the comparison we adopt here the best-reply representation of equilibrium actions; see equation (5).
46
We maintain however the assumption of symmetric neighborhoods, an assumption which, as is the case for linearity, renders identification harder: see Bramoull´e, Djebbari and Fortin (2009) for a study of the identification power of observable asymmetric neighborhoods.
47
See also Cabral (1990) for an early discussion of these issues and Young (2009); see Blume, Brock, Durlauf, Ioannides (2011) for an up to date survey.
